The problem asks us to find the value of $x$, given the points $A(x, 5)$ and $B(-3, 1)$, with the length of $AB = 5$.

Solution:

The formula for the distance between two points is: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substitute the points $A(x, 5)$ and $B(-3, 1)$ into the formula: $5 = \sqrt{((-3) - x)^2 + (1 - 5)^2}$

Simplify: $5 = \sqrt{(x + 3)^2 + (-4)^2}$ $5 = \sqrt{(x + 3)^2 + 16}$

Square both sides to eliminate the square root: $25 = (x + 3)^2 + 16$

Simplify: $25 - 16 = (x + 3)^2$ $9 = (x + 3)^2$

Take the square root of both sides: $x + 3 = \pm 3$

Solve for $x$:

1. $x + 3 = 3$ ⟹ $x = 0$
2. $x + 3 = -3$ ⟹ $x = -6$

Final Answer:

The possible values of $x$ are: $x = 0 \quad \text{or} \quad x = -6$

Would you like further details or have any additional questions?

5 Related Questions:

1. What is the formula for the midpoint of two points?
2. How does squaring both sides of an equation affect the solution?
3. Can there be multiple solutions to a distance equation? Why?
4. How would the solution change if the length of $AB$ were different?
5. How can this problem be represented graphically?

Tip:

Always verify your solutions by substituting back into the original equation to ensure they satisfy the given conditions.